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Technical Commission VII

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In general, MAP estimator is computationally efficient and is
not sensitive to irregular sampling. However, it obeys the
Rayleigh resolution limit, i.e. it has almost no super-resolution
32.2 Nonlinear Least Squares (NLS)
Assuming the presence of K scatterers inside a pixel with
elevations of s=[s, 8a, ae the under-determined
system model (1) reduces to the following over-determined
$us =(R*(s)R(s)) R"(s)g 3)
For Gaussian white noise, NLS is identical to the maximum
likelihood estimator (MLE). It is therefore theoretically the best
estimator for our application if and only if the data closely
agree with the assumed model. Due to the large computational
effort to the multi-dimensional search, the NP-hard NLS is not
recommended for practical data processing.
The "Scale-down by L1 norm Minimization, Model selection,
and Estimation Reconstruction" (SLIMMER, pronounced
"slimmer") algorithm is a spectral estimator firstly proposed in
(Zhu and Bamler, 2010b). It consists of three main steps: 1) a
dimensionality scale-down by L1 norm minimization, 2) model
selection and 3) linear parameter estimation. In case there is no
prior knowledge about the number of scatterers inside the pixel
and in the presence of measurement noise, the sparse
reconstruction of (1) is give by the following L1- L2 norm

adr) @
The L;-L, norm minimization step shrinks R dramatically and
gives a first sparse estimate of y. Due to the fact the following
two effects (Zhu, 2011): 1) for our application, RIP and
incoherence are violated for several reasons 2) - The L1 norm
approximation of the NP-hard LO norm regularization
introduces amplitude bias, This estimate may still contain the
outliers. Therefore, a further model order selection and
parameter estimation step are followed to refine the sparse
estimates obtained from (5).
SLIMMER offers an aesthetic non-parametric realization of the
NP-hard NLS estimator. As an efficient estimator, it is
demonstrated to provide a super-resolution capability reaches
the fundamental bounds of all spectral estimators.
Younes = 918 min {lg -Ry
3.3 Tomographic SAR Reconstruction with Colored Noise
Based on the discussion in Section 2.2, i.e. the different data
quality possessed by repeat- and single-pass data can be
therefore characterized by the noise variance matrix C,, , the
aforementioned estimators are extended to the colored noise
case as following:
e MAP Estimator
f,» -(R" C2 R41) R*Czg (5)
* Nonlinear Least Squares (NLS)
faus-(R'(sCZR(s) R*()C2g ©
hu cargmin((g-Ry) C?(g-Rr)e2.|rh] — C
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
3.4 Cramér-Rao Lower Bound of the Elevation Estimates
Under colored noise, the Cramér-Rao Lower Bound (CRLB) of
the elevation estimates in presence of only a single scatterer is
given by:
S mr E (8)
4r 42 |V SNR, 6,
where SNR, stands for the signal-to-noise ratio (SNR) of the
n? acquisition (n 21,...,N ). 6, is the re-weighted standard
deviation of the baseline distribution:
Y.SNR,b; (Y,SNR,b
Ge | Az — (9)
Compared to the CRLB under Gaussian white noise 0 (i.e. all
data possesses the same SNR ), the N- SNR dependent factor is
replaced by D SNR, to account for variant data quality and the
standard deviation of the baseline distribution o, is
accordingly replaced by the &, depends on the sampling
position b, re-weighted according to the data quality SNR, . It
tells that, in case of adding several high quality TanDEM-X
data to an existing TerraSAR-X stack, the elevation estimation
accuracy improvement depends not only on the number of
TanDEM-X images, but also the corresponding baseline
distribution. Le. widely spread baselines are preferred.
Considering a mixed stack consisting of N, images with a
higher SNR of SNR, and N, images with a lower SNR of
SNR, , let’s assume the standard deviation of the baseline
distribution for each sub-stack are o,, and o,, , eq. (8) can be
approximated by:
o EE (10)
YW ve,
i=12 (11)
"4n A2 JN, SNR, o,,
A reasonable data quality assumption for a mixed repeat- and
single-pass data stack is that we have a mixed stack consisting
of N, images with a higher SNR of SNR, and N, images
with a lower SNR of SNR, .
The data is simulated using the regularly distributed elevation
aperture shown in Fig.5 (25 images, elevation resolution
p, = 40.5m ) with the following two cases:
e TSX: multi-pass data stack case, i.e. 25 images with
e TDX: mixed stack, 20 images with SNR of 5dB
(elevation aperture positions marked as black) and 5
images with SNR=20dB (marked as green).
Fig.6 shows comparison of the reflectivity profiles along
elevation direction reconstructed by MAP (blue) and
SLIMMER (red) using the data stack of TSX and TDX cases
(right). The true elevations of the two scatterers are 10 and 25
meter, respectively, an i.e. elevation distance is 0.4 times of the
Rayleigh resolution limit. It is obvious to see the better sidelobe
suppression for the non-parametric estimator and better super-